
theorem Th3:
  for S,T being TopStruct, f being Function of S,T holds
  f is being_homeomorphism implies f" is being_homeomorphism
proof
  let S, T be TopStruct;
  let f be Function of S, T;
  assume
A1: f is being_homeomorphism;
  then
A2: dom f = [#]S & rng f = [#]T & f is one-to-one &
  f is continuous & f" is continuous by TOPS_2:def 5;
  per cases;
  suppose S is non empty & T is non empty;
    hence f" is being_homeomorphism by A1,TOPS_2:56;
  end;
  suppose A3: S is empty or T is empty;
A4: f = {}
    proof
      per cases by A3;
      suppose S is empty; hence thesis; end;
      suppose T is empty; hence thesis; end;
    end;
    reconsider g = f as onto one-to-one PartFunc of {},{} by A4,FUNCTOR0:1;
A5: f" = g" by A2;
A6: dom(f") = [#]T & rng(f") = [#]S by A2,A4;
    reconsider g1 = f" as onto one-to-one PartFunc of {},{} by A5;
    (f")" = g1" by A2,A4;
    hence thesis by A4,A6,A2,TOPS_2:def 5;
  end;
end;
